I was reading Sean Carroll book "Space-Time and geometry", in the appendix B he derives the energy momentum conservation from the diffeomorphism invariance of the action, however I don't understand a step in the derivation.

I will put some context before asking the question.

He starts with the action for matter fields $S_{m}$ (in the context of general relativity), and takes de variation of the action

$$ \delta S_{m}=\int d^{4}x \frac{\delta S_{m}}{ \delta g_{ab}}\delta g_{ab}+\int d^{4}x \frac{\delta S_{m}}{ \delta \psi} \psi$$ The matter field equations tell us that $\delta S_{m}/\delta \psi=0$, then $$\delta S_{m}=\int d^{4}x \frac{\delta S_{m}}{ \delta g_{ab}}\delta g_{ab}= \int d^{4}x \sqrt{-g} T^{\mu\nu}\nabla_{\mu}\xi_{\nu}$$ Where I have used the definition of Lie derivative (I use the letter $\zeta$ to denote the lie derivative) of the metric $$ \delta_{\xi}g_{\mu\nu}=\zeta_{\xi}g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$$ And the definition of energy momentum tensor $$T_{ab}=\frac{-2}{\sqrt{-g}}\frac{\delta S_{m}}{\delta g_{ab}}$$ Next, in the equation (B.25) Sean Carroll makes $$\delta S_{m}=\int d^{4}x \sqrt{-g} T^{\mu\nu}\nabla_{\mu}\xi_{\nu}\\ =-\int d^{4}x \sqrt{-g} \xi_{\nu}\nabla_{\mu} T^{\mu\nu}$$ I don't understand how he can change the covariant derivative, it looks like some kind of integration by parts, but I don't see what happened to the boundary term.


1 Answer 1


Boundary terms are always assumed zero in these considerations. After all, non-zero would mean that some energy-momentum is incoming/outgoing. Inside can't be conserved if that's the case.

You can of course write a more comprehensive conservation law by including how much comes in/out (integration over the boundary), but this is not the point here.

  • $\begingroup$ I read that the people assume that $\xi^{a}$ has compact support. $\endgroup$
    – Nothing
    Commented May 15, 2020 at 21:45
  • $\begingroup$ This is not the correct answer. Even if the stress-energy tensor vanishes at spatial/null infinity (i.e. no incoming/outgoing flux) there is still nonvanishing stress-energy tensor at future/past timelike infinity, which doesn't mean you are loosing/getting extra energy, just that your energy distribution must have come from somewhere and ended up somewhere. Indeed, the way to get the result is to assume that $\xi$ has compact support, which is always correct since we assume diffeomorphism invariance under arbitrary vector fields. $\endgroup$ Commented Apr 4 at 18:27

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